The equation of the parabola given is y = 112x². This is a standard form of a parabola that opens upwards. To find the focus and directrix, we first need to rewrite the equation in the standard form of a parabola.
The standard form for a parabola that opens upwards is:
y = 4p(x – h)² + k
In this equation:
- (h, k) is the vertex of the parabola,
- p is the distance from the vertex to the focus.
Comparing y = 112x² with the standard form, we can see that:
- h = 0
- k = 0
- 4p = 112
From the equation 4p = 112, we can solve for p:
p = 112 / 4 = 28
This means that the focus is located at a distance of 28 units above the vertex, which is at (0, 0).
Therefore, the coordinates of the focus are:
(0, 28)
Next, we need to find the equation of the directrix, which is a horizontal line located p units below the vertex.
Since the vertex is at (0, 0) and p = 28, the directrix is:
y = 0 – 28 = -28
In conclusion:
- Focus: (0, 28)
- Directrix: y = -28